Question

If $\sin \theta +\cos \theta =p$ and ${\sin }^3\theta +{\cos }^3\theta =q$ then $p\left(p^2-3\right)$ is equal to

• A. q
• B. 2 q
• C. -q
• D. -2 q

Answer 1

Answer: (D)

Given, $\sin \theta +\cos \theta =p$ ...(i)
and ${\sin }^3\theta +{\cos }^3\theta =q$ ...(ii)
$\Rightarrow (\sin \theta +\cos \theta )$
$\left({\sin }^2\theta -\sin \theta \cdot \cos \theta +{\cos }^2\theta \right)=q$
$\Rightarrow p(1-\sin \theta \cdot \cos \theta )=q$
[From Eq. (i) and ${\sin }^2\theta +{\cos }^2\theta =1]$
$\Rightarrow 1-\sin \theta \cdot \cos \theta =\frac{q}{p}$
$\Rightarrow \sin \theta \cdot \cos \theta =1-\frac{q}{p}$ ...(iii)
On squaring both sides of Eq. (i), we get
${\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cdot \cos \theta =p^2$
$\Rightarrow 1+2\left(1-\frac{q}{p}\right)=p^2$ [from Eq. (iii)]
$\Rightarrow p+2(p-q)=p^3$
$\Rightarrow 3p-2q=p^3$
$\Rightarrow p^3-3p=-2q$
$\Rightarrow p\left(p^2-3\right)=-2q$